Counting cospherical tuples in lattice cubes
For integers $d \ge 3$ and $n \ge 1$, let $S(n,d)$ be the number of ordered $(d+2)$-tuples of pairwise distinct points of $\{1, \ldots, n\}^d$ that lie on some Euclidean sphere $$\{x \in \mathbb{R}^d : \|x - a\|_2 = R\}, \qquad a \in \mathbb{R}^d, \quad R > 0.$$ A tuple is counted once even if several spheres contain it, and hyperplanes are not counted as spheres. Ghosal, Goenka and Keevash proved that $$n^{d^2+d-2} \lesssim S(n,d) \lesssim n^{d^2+2d-4+c/\log\log n} + n^{d^2+d}\log n$$ for every $d \ge 3$, with a constant $c$, and that the number of cyclic quadrilaterals with vertices in $\{1, \ldots, n\}^2$ is $\gamma n^5 + O(n^{4 + 18/29 + \varepsilon})$ for every $\varepsilon > 0$, with an explicit constant $\gamma$.
Problem. Is there a constant $C_d$, independent of $n$, such that $$S(n,d) \le C_d\, n^{d^2+d-1}$$ for all $n$?
Source: Ghosal, Goenka and Keevash (2026), Conjecture 5.1, for all d ≥ 2 (they prove d = 2); stated for d ≥ 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- MG, NT
- Keywords
- AIM Open Applied Problems, cospherical points, incidence geometry, lattice points
References
@article{ghosal2026,
author = {Ghosal, Anubhab and Goenka, Ritesh and Keevash, Peter},
title = {On subsets of lattice cubes avoiding affine and spherical degeneracies},
journal = {Discrete Comput. Geom.},
volume = {76},
number = {3},
year = {2026},
pages = {1886--1911},
doi = {10.1007/s00454-026-00853-7},
eprint = {2509.06935},
archivePrefix = {arXiv},
}
@misc{aim-cospherical-lattice-tuple-count,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Counting cospherical tuples in lattice cubes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/611-cospherical-lattice-tuple-count.md},
note = {Entry 611 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00854, "Counting cospherical tuples in lattice cubes", clopen.solutions.